Problem 11 - Algebra 2 and Trigonometry

Question

Is arctan \(1=220^{\circ}\) a true statement? Justify your answer. \(y=\arctan (-1)\)

Step-by-Step Solution

Answer

No, \(\arctan(1) = 45^\circ\), not \(220^\circ\). Similarly, \(\arctan(-1) = -45^\circ\).

Step 1: Understand the arctan function

The function \(y = \arctan(x)\) represents the angle \(y\) such that \(\tan(y) = x\). It typically returns values within the interval \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians, or \(-90^\circ\) to \(90^\circ\) if measured in degrees.

Step 2: Recognize the given statement

The problem statement claims \(\arctan(1) = 220^\circ\). However, since \(\arctan\) only returns angles between \(-90^\circ\) and \(90^\circ\), \(220^\circ\) is outside this range. This means \(\arctan(1) = 220^\circ\) is not possible.

Step 3: Calculate \(\arctan(1)\)

The value \(\arctan(1)\) is the angle whose tangent is 1. The basic angle in this range where \(\tan(\theta) = 1\) is \(45^\circ\) or \(\frac{\pi}{4}\) radians, not \(220^\circ\). Thus, \(\arctan(1) = 45^\circ\).

Step 4: Calculate \(\arctan(-1)\)

For the angle with a tangent of -1, we need to find the angle in the range \(-90^\circ\) to \(90^\circ\). This angle is \(-45^\circ\) or \(-\frac{\pi}{4}\) radians because \(\tan(-45^\circ) = -1\). Thus, \(\arctan(-1) = -45^\circ\) or \(-\frac{\pi}{4}\) radians.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles when we know the trigonometric ratios. In simpler terms, they are the reverse of the regular trigonometric functions like sine, cosine, and tangent. Instead of starting with an angle to find these ratios, we start with the ratio and find the angle. Some common inverse trigonometric functions include

• Arcsin, which gives the angle for a given sine value.
• Arccos, which helps find the angle for a given cosine value.
• Arctan, which returns the angle when given a tangent value.

In our example with arctan, if you know the tangent of the angle is 1, the inverse trigonometric function, arctan, will return the angle, which is 45 degrees. This process is very useful because it allows us to work backwards when solving problems related to angle measurements.

Angle Measurement

Angles can be measured in two ways: degrees and radians. Understanding both systems is important when working with trigonometry. Here are some key points:

• Degrees are more commonly used in everyday applications, such as geometry problems in school. A full circle is 360 degrees.
• Radians are used more frequently in higher mathematics, especially calculus. In radians, a full circle is described as \(2\pi\) rad. This system relates to the arc length of a circle.

In trigonometry, certain functions, like arctan, have specific ranges where they return results. For arctan, the angles are between \(-90^{\circ}\) and \(90^{\circ}\), or equivalently \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians.
When you apply arctan to a value like 1, the answer will fit inside this range, giving us \(45^{\circ}\) or \(\frac{\pi}{4}\) radians, but definitely not \(220^{\circ}\) as claimed in the failed statement.

Tangent Function

The tangent function is one of the key trigonometric functions, often abbreviated as \(\tan\). It relates an angle in a right triangle to the ratio of the side opposite the angle to the adjacent side. Mathematically, if you have a right triangle,

• Opposite side: the side opposite the angle in question
• Adjacent side: the side next to the angle in question (excluding the hypotenuse)

Then, the tangent of the angle \(\theta\) is given by:\[\tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\]
When applied, this function not only helps in solving triangles but also aids in understanding angles beyond those of a triangle through its periodicity.
With inverse trigonometric functions like arctan, you can find the original angle that has the given tangent ratio. For instance, an angle whose tangent is 1 will return \(45^{\circ}\) on applying arctan, comfortably inside its defined range of \(-90^{\circ}\) to \(90^{\circ}\). This highlights why \(\arctan(1) = 220^{\circ}\) is incorrect, as \(220^{\circ}\) can't possess the tangent value of 1 within the fundamental range provided by the arctan function.